Sure! Let's work through this problem step-by-step.
We are given two points [tex]\((10, 9)\)[/tex] and [tex]\((5, 12)\)[/tex], and we need to find the equation of the line that passes through these points. The first step is to find the slope [tex]\(m\)[/tex] of the line.
The formula for the slope [tex]\(m\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in the coordinates of the points [tex]\((x_1, y_1) = (10, 9)\)[/tex] and [tex]\((x_2, y_2) = (5, 12)\)[/tex]:
[tex]\[ m = \frac{12 - 9}{5 - 10} = \frac{3}{-5} = -0.6 \][/tex]
So, the slope [tex]\(m\)[/tex] is:
[tex]\[ m = -0.6 \][/tex]
Next, we need to find the y-intercept [tex]\(b\)[/tex] of the line. The y-intercept can be found using the slope-intercept form of a linear equation, which is:
[tex]\[ y = mx + b \][/tex]
We can rearrange this formula to solve for [tex]\(b\)[/tex]:
[tex]\[ b = y - mx \][/tex]
Using one of the given points, let's use [tex]\((10, 9)\)[/tex] and the slope [tex]\(m = -0.6\)[/tex]:
[tex]\[ b = 9 - (-0.6) \times 10 \][/tex]
[tex]\[ b = 9 + 6 \][/tex]
[tex]\[ b = 15 \][/tex]
So, the y-intercept [tex]\(b\)[/tex] is:
[tex]\[ b = 15.0 \][/tex]
Thus, the answers are:
[tex]\[ \boxed{-0.6} \][/tex]
[tex]\[ \boxed{15.0} \][/tex]
